reviewsol - ODE, Review for Exam 1 June 9, 2008 1. Consider...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ODE, Review for Exam 1 June 9, 2008 1. Consider the autonomous equation x = x 3- x . (a) Sketch the phase line (the x axis) and the corresponding direction field on the plane. (b) Sketch graphs of some solutions. Be sure to include at least one solution with values in each interval above, below, and between critical points. (c) Suppose x ( t ) is a nonconstant solution to the above equation. If x 00 (2) = 0 what is x (2)? (c) There are two possibilities: x = 1 / 3. 2. Consider the equation x 00 + 2 x + 2 x = e- t cos 2 t . (a) Find a particular solution. (b) Find the general solution. (a) Using variation of parameters, I find a particuar solution y p ( x ) = (- cos t + 2 3 cos 3 t )( e- t cos t )+ (sin t- 2 3 sin 3 t )( e- t sin t ). (b) y = c 1 e- t cos t + c 2 e- t sin t + (- cos t + 2 3 cos 3 t )( e- t cos t ) + (sin t- 2 3 sin 3 t )( e- t sin t ). 3. (a) If e- 2 t + 2 e- t solves x 00 + cx + kx = 0 what are c, k ? (b) Same if te- t instead. (a) c = 3, k = 2 (b) c = 2, k = 1 4. Solve y 00- 4 y = e 2 t . y = c 1 e 2 t + c 2 e- 2 t + 1 4 te 2 t . 5. Consider the equation y = ty 2 ....
View Full Document

Page1 / 3

reviewsol - ODE, Review for Exam 1 June 9, 2008 1. Consider...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online